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Journal of Physics C Solid State Physics

Position space studies of the pure and dilute quantum Ising model. I. One-dimensional chain

G. Kamieniarz, L. S. Campana, Alvaro Caramico D’Auria, Ugo Esposito

Published Mar 30, 1987• 7 citations
Publisher

Abstract

A block-scaling method describing in a unified way quantum, thermal and random effects in the one-dimensional Ising model with a transverse field is presented. Reliable results have been obtained using a free-energy preserving mapping in a suitable cumulant-like approximation. At zero temperature a number of exact results characterising the quantum and percolative critical behaviour of the pure and dilute chain is recovered.

DOI
10.1088/0022-3719/20/9/023

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Journal of Physics A Mathematical and General•Dec 1, 1980•23 citations
A block-scaling method describing in a unified way quantum, thermal and random effects in the one-dimensional Ising model with a transverse field is presented. Reliable results have been obtained using a free-energy preserving mapping in a suitable cumulant-like approximation. At zero temperature a number of exact results characterising the quantum and percolative critical behaviour of the pure and dilute chain is recovered.
For pt.III see ibid., vol.20, p.1841 (1987). The global real-space renormalisation group approach is applied in two and three dimensions to study the pure and diluted transverse Ising model at finite temperatures. A phase diagram in the temperature-field plane is presented for the pure quantum model on the honeycomb, triangular and square lattices in the first-order cumulant expansion. An improvement of the classical Ising fixed-point data in two and three dimensions is considered by having recourse to larger cells and the second-order cumulant expansion. The zero-temperature percolative behaviour of the bond-diluted system is recovered from the fixed-point data analysis. Some shortcomings and relations to other approaches are also discussed.
A zero-temperature real space renormalisation group block method is applied to the random quantum Ising model with a transverse field on the planar honeycomb and square lattices. For the bound diluted system the magnetisation and the separation of the ground-state energy level (in the paramagnetic phase) are presented for several bond concentrations p. The critical exponents extracted both from the fixed points and from direct numerical computations preserve some scaling relations, and the critical curve displays a characteristic discontinuity at the percolation concentration. For the McCoy and Wu distribution (1968) the random fields and bonds are found to introduce a strong relevant disorder. The order parameter still falls off continuously to zero for well defined values of the parameters but a new fixed point yields a slight change in the critical exponents.
The random Ising model in a transverse field in one dimension with Hamiltonian H=- Sigma ( Gamma i S i z +J i S i x S i+1 x ) is studied at T=O from a real-space renormalisation group block method which preserves duality transformations. The ground state magnetisation and the ground state energy are determined for random distributions P(J)(P( Gamma ))=N J J NJ-1 /J 0 N(J) for 0<J<J 0 and P(J)=0 for J>J O . A new critical behaviour corresponding to a new fixed point takes place in the presence of disorder. The crossover exponent describing the departure from the pure system behaviour is calculated. The second derivative of the ground state energy delta 2 E/ delta Gamma 2 which diverges logarithmically for the pure system is rounded in the presence of disorder but a sharp transition field still exists where the magnetisation goes to zero with an exponent beta about twice as large as the pure system exponent. Comparison is made with the analytical results of McCoy and Wu (1969) for the classical equivalent random-striped Ising 2D model.